Is this progression functional? 4mim/b3 1/3 1dom7 4maj. Numbers to the right of the slash are left hand (bass) single notes that accompany their respective chords to the left of the slash.
If I understand your notation the first chord is a minor on the 4th degree of the Key, I would use IV-. Then followed by a I first inversion, a I7 then a IV maj7. This is a very useful progression. The IV- resolves to I, though I'd expect a passage to start on I in most cases. The I --> I7 is the perfect set up to modulate to the IV.
I often teach the following progression on guitar to my students and I thing it is similar functionally to the one you posted.
I --> I7 --> IV --> IV-
This moves from I to IV but includes proper cycle extensions that create resolution to the I or the IV.
What do you mean by 'functional? Generally the term means 'it works'. In music, something 'works' for one person, but may not 'work' for another.
Trying to translate your chords into a form that is better known, in key C: Fm/A♭, C/E, C7, F. Yes, of course it works! Even sounds quite good! But where is it in the piece? Starting with Fm? That sounds like it's modulated from Fm into Fmajor. Nothing wrong with that. It could depend on what key the piece is in, and what preceded it and followed.
But the answer is yes - it's functional. And if you think it sounds good, that counts for even more!
As you probably gathered (from ggcg's answer and mine), there are better known ways in which to write chord sequences - using RN, or NNS are the most used.